Integrand size = 25, antiderivative size = 442 \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a-b x^2}} \, dx=-\frac {2 d^2 \sqrt {c+d x} \sqrt {a-b x^2}}{3 b}-\frac {14 \sqrt {a} c d \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} d \left (2 b c^2+a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 c^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/3*d^2*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b-14/3*a^(1/2)*c*d*(d*x+c)^(1/2)*( 1-b*x^2/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2) *(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/(b^(1/2)*(d*x+c)/(b^(1/2 )*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-2/3*a^(1/2)*d*(a*d^2+2*b*c^2)*(b^(1 /2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/2)*EllipticF(1/2*( 1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d ))^(1/2))/b^(3/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-2*c^3*(b^(1/2)*(d*x+c)/(b ^(1/2)*c+a^(1/2)*d))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)* x/a^(1/2))^(1/2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2) )/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.39 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.55 \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a-b x^2}} \, dx=\frac {2 d^2 \sqrt {a-b x^2} \left (-c-d x+\frac {-7 b c^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+7 a c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+14 b c^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)-7 b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2+7 i \sqrt {b} c \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \left (6 b c^2-7 \sqrt {a} \sqrt {b} c d+a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-3 i b c^2 \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{3 b \sqrt {c+d x}} \] Input:
Integrate[(c + d*x)^(5/2)/(x*Sqrt[a - b*x^2]),x]
Output:
(2*d^2*Sqrt[a - b*x^2]*(-c - d*x + (-7*b*c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b] ] + 7*a*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 14*b*c^2*Sqrt[-c + (Sqrt[a] *d)/Sqrt[b]]*(c + d*x) - 7*b*c*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + (7*I)*Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/( c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*E llipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b] *c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*(6*b*c^2 - 7*Sqrt[a]*Sqrt[b]* c*d + a*d^2)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d) /Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - (3*I)*b*c^2*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[- (((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(Sqrt[ b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sq rt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d^2*Sqrt[ -c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(3*b*Sqrt[c + d*x])
Time = 0.95 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {634, 633, 632, 186, 413, 412, 2185, 27, 600, 509, 508, 327, 512, 511, 321}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x \sqrt {a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 634 |
| \(\displaystyle c^3 \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-\int \frac {-x^2 d^3-3 c x d^2-3 c^2 d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \frac {c^3 \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\int \frac {-x^2 d^3-3 c x d^2-3 c^2 d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {c^3 \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{x \sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+d x}}dx}{\sqrt {a-b x^2}}-\int \frac {-x^2 d^3-3 c x d^2-3 c^2 d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle -\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}-\int \frac {-x^2 d^3-3 c x d^2-3 c^2 d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle -\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\int \frac {-x^2 d^3-3 c x d^2-3 c^2 d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\int \frac {-x^2 d^3-3 c x d^2-3 c^2 d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {2 \int \frac {d^3 \left (9 b c^2+7 b d x c+a d^2\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int \frac {9 b c^2+7 b d x c+a d^2}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {d \left (\left (a d^2+2 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+7 b c \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx\right )}{3 b}-\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {d \left (\left (a d^2+2 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {7 b c \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}\right )}{3 b}-\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {d \left (\left (a d^2+2 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {14 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b}-\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {d \left (\left (a d^2+2 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {14 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b}-\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {d \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (a d^2+2 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {14 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b}-\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {d \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (a d^2+2 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {14 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b}-\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 c^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}+\frac {d \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (a d^2+2 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {14 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b}-\frac {2 d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\) |
Input:
Int[(c + d*x)^(5/2)/(x*Sqrt[a - b*x^2]),x]
Output:
(-2*d^2*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(3*b) + (d*((-14*Sqrt[a]*Sqrt[b]*c* Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sq rt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[(Sqrt[b]*(c + d*x ))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*(2*b*c^2 + a*d^2 )*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*El lipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/ Sqrt[a] + d)])/(Sqrt[b]*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(3*b) - (2*c^3*Sq rt[1 - (b*x^2)/a]*Sqrt[1 - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/(Sqrt[b]* c + Sqrt[a]*d)]*EllipticPi[2, ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2] ], (2*Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[a - b*x^2]*Sqrt[c + (Sqrt [a]*d)/Sqrt[b] - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/Sqrt[b]])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 + b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))^(n_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[c^(n + 1/2) Int[1/(x*Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] - Int[( 1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(c^(n + 1/2) - (c + d*x)^(n + 1/2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n - 1/2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(817\) vs. \(2(355)=710\).
Time = 1.72 (sec) , antiderivative size = 818, normalized size of antiderivative = 1.85
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 d^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b}+\frac {2 \left (3 c^{2} d +\frac {a \,d^{3}}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {14 c \,d^{2} \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{3 \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {2 c^{2} \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, d \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(818\) |
| default | \(\text {Expression too large to display}\) | \(1253\) |
Input:
int((d*x+c)^(5/2)/x/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2/3*d^2/b*(-b* d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(3*c^2*d+1/3*a/b*d^3)*(c/d-1/b*(a*b)^(1/2 ))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a *b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d *x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1 /2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+14/3*c*d^2*(c/d -1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2 ))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2 )))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2))*Ellip ticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1 /b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^ (1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))-2*c^ 2*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b )^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b )^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*d*EllipticPi(((x+c/d)/( c/d-1/b*(a*b)^(1/2)))^(1/2),-(-c/d+1/b*(a*b)^(1/2))/c*d,((-c/d+1/b*(a*b)^( 1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
Timed out. \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a-b x^2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(5/2)/x/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a-b x^2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x \sqrt {a - b x^{2}}}\, dx \] Input:
integrate((d*x+c)**(5/2)/x/(-b*x**2+a)**(1/2),x)
Output:
Integral((c + d*x)**(5/2)/(x*sqrt(a - b*x**2)), x)
\[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}}}{\sqrt {-b x^{2} + a} x} \,d x } \] Input:
integrate((d*x+c)^(5/2)/x/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(5/2)/(sqrt(-b*x^2 + a)*x), x)
\[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}}}{\sqrt {-b x^{2} + a} x} \,d x } \] Input:
integrate((d*x+c)^(5/2)/x/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(5/2)/(sqrt(-b*x^2 + a)*x), x)
Timed out. \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a-b x^2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x\,\sqrt {a-b\,x^2}} \,d x \] Input:
int((c + d*x)^(5/2)/(x*(a - b*x^2)^(1/2)),x)
Output:
int((c + d*x)^(5/2)/(x*(a - b*x^2)^(1/2)), x)
\[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a-b x^2}} \, dx=\frac {-6 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, d^{2}-7 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b \,d^{3}+2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{4}-b c \,x^{3}+a d \,x^{2}+a c x}d x \right ) b \,c^{3}+3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,d^{3}+6 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b \,c^{2} d}{2 b} \] Input:
int((d*x+c)^(5/2)/x/(-b*x^2+a)^(1/2),x)
Output:
( - 6*sqrt(c + d*x)*sqrt(a - b*x**2)*d**2 - 7*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b*d**3 + 2*int((sqrt( c + d*x)*sqrt(a - b*x**2))/(a*c*x + a*d*x**2 - b*c*x**3 - b*d*x**4),x)*b*c **3 + 3*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d *x**3),x)*a*d**3 + 6*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b *c*x**2 - b*d*x**3),x)*b*c**2*d)/(2*b)