\(\int \frac {(c+d x)^{3/2}}{(e x)^{11/2} \sqrt {b c^2-b d^2 x^2}} \, dx\) [1000]

Optimal result

Integrand size = 36, antiderivative size = 248 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{11/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {2 \sqrt {b c^2-b d^2 x^2}}{9 b e (e x)^{9/2} \sqrt {c+d x}}-\frac {34 d \sqrt {b c^2-b d^2 x^2}}{63 b c e^2 (e x)^{7/2} \sqrt {c+d x}}-\frac {68 d^2 \sqrt {b c^2-b d^2 x^2}}{105 b c^2 e^3 (e x)^{5/2} \sqrt {c+d x}}-\frac {272 d^3 \sqrt {b c^2-b d^2 x^2}}{315 b c^3 e^4 (e x)^{3/2} \sqrt {c+d x}}-\frac {544 d^4 \sqrt {b c^2-b d^2 x^2}}{315 b c^4 e^5 \sqrt {e x} \sqrt {c+d x}} \] Output:

-2/9*(-b*d^2*x^2+b*c^2)^(1/2)/b/e/(e*x)^(9/2)/(d*x+c)^(1/2)-34/63*d*(-b*d^ 
2*x^2+b*c^2)^(1/2)/b/c/e^2/(e*x)^(7/2)/(d*x+c)^(1/2)-68/105*d^2*(-b*d^2*x^ 
2+b*c^2)^(1/2)/b/c^2/e^3/(e*x)^(5/2)/(d*x+c)^(1/2)-272/315*d^3*(-b*d^2*x^2 
+b*c^2)^(1/2)/b/c^3/e^4/(e*x)^(3/2)/(d*x+c)^(1/2)-544/315*d^4*(-b*d^2*x^2+ 
b*c^2)^(1/2)/b/c^4/e^5/(e*x)^(1/2)/(d*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 6.61 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.37 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{11/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {2 \sqrt {e x} \sqrt {b \left (c^2-d^2 x^2\right )} \left (35 c^4+85 c^3 d x+102 c^2 d^2 x^2+136 c d^3 x^3+272 d^4 x^4\right )}{315 b c^4 e^6 x^5 \sqrt {c+d x}} \] Input:

Integrate[(c + d*x)^(3/2)/((e*x)^(11/2)*Sqrt[b*c^2 - b*d^2*x^2]),x]
 

Output:

(-2*Sqrt[e*x]*Sqrt[b*(c^2 - d^2*x^2)]*(35*c^4 + 85*c^3*d*x + 102*c^2*d^2*x 
^2 + 136*c*d^3*x^3 + 272*d^4*x^4))/(315*b*c^4*e^6*x^5*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {580, 579, 579, 579, 579}

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.

\(\displaystyle \int \frac {(c+d x)^{3/2}}{(e x)^{11/2} \sqrt {b c^2-b d^2 x^2}} \, dx\)

\(\Big \downarrow \) 580

\(\displaystyle \frac {17 d \int \frac {\sqrt {c+d x}}{(e x)^{9/2} \sqrt {b c^2-b d^2 x^2}}dx}{9 e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{9 b e (e x)^{9/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 579

\(\displaystyle \frac {17 d \left (\frac {6 d \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \sqrt {b c^2-b d^2 x^2}}dx}{7 c e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{7 b c e (e x)^{7/2} \sqrt {c+d x}}\right )}{9 e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{9 b e (e x)^{9/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 579

\(\displaystyle \frac {17 d \left (\frac {6 d \left (\frac {4 d \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \sqrt {b c^2-b d^2 x^2}}dx}{5 c e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{5 b c e (e x)^{5/2} \sqrt {c+d x}}\right )}{7 c e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{7 b c e (e x)^{7/2} \sqrt {c+d x}}\right )}{9 e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{9 b e (e x)^{9/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 579

\(\displaystyle \frac {17 d \left (\frac {6 d \left (\frac {4 d \left (\frac {2 d \int \frac {\sqrt {c+d x}}{(e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}dx}{3 c e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{3 b c e (e x)^{3/2} \sqrt {c+d x}}\right )}{5 c e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{5 b c e (e x)^{5/2} \sqrt {c+d x}}\right )}{7 c e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{7 b c e (e x)^{7/2} \sqrt {c+d x}}\right )}{9 e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{9 b e (e x)^{9/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 579

\(\displaystyle \frac {17 d \left (\frac {6 d \left (\frac {4 d \left (-\frac {4 d \sqrt {b c^2-b d^2 x^2}}{3 b c^2 e^2 \sqrt {e x} \sqrt {c+d x}}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{3 b c e (e x)^{3/2} \sqrt {c+d x}}\right )}{5 c e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{5 b c e (e x)^{5/2} \sqrt {c+d x}}\right )}{7 c e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{7 b c e (e x)^{7/2} \sqrt {c+d x}}\right )}{9 e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{9 b e (e x)^{9/2} \sqrt {c+d x}}\)

Input:

Int[(c + d*x)^(3/2)/((e*x)^(11/2)*Sqrt[b*c^2 - b*d^2*x^2]),x]
 

Output:

(-2*Sqrt[b*c^2 - b*d^2*x^2])/(9*b*e*(e*x)^(9/2)*Sqrt[c + d*x]) + (17*d*((- 
2*Sqrt[b*c^2 - b*d^2*x^2])/(7*b*c*e*(e*x)^(7/2)*Sqrt[c + d*x]) + (6*d*((-2 
*Sqrt[b*c^2 - b*d^2*x^2])/(5*b*c*e*(e*x)^(5/2)*Sqrt[c + d*x]) + (4*d*((-2* 
Sqrt[b*c^2 - b*d^2*x^2])/(3*b*c*e*(e*x)^(3/2)*Sqrt[c + d*x]) - (4*d*Sqrt[b 
*c^2 - b*d^2*x^2])/(3*b*c^2*e^2*Sqrt[e*x]*Sqrt[c + d*x])))/(5*c*e)))/(7*c* 
e)))/(9*e)
 

Defintions of rubi rules used

rule 579

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 
1)/(b*c*e*(m + 1))), x] - Simp[d*((n - m - 2)/(c*e*(m + 1)))   Int[(e*x)^(m 
 + 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] 
&& EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0] && LtQ[m, -1] && (IntegerQ[2*p] | 
| IntegerQ[m])
 

rule 580

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 2)*((a + b*x^2)^(p + 
1)/(b*e*(m + 1))), x] + Simp[d*((2*m + p + 3)/(e*(m + 1)))   Int[(e*x)^(m + 
 1)*(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, 
 x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p - 1, 0] && LtQ[m, -1] && IntegerQ 
[p + 1/2]
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.33

method	result	size
gosper	\(-\frac {2 \left (-d x +c \right ) x \left (272 d^{4} x^{4}+136 c \,d^{3} x^{3}+102 d^{2} c^{2} x^{2}+85 c^{3} d x +35 c^{4}\right ) \sqrt {d x +c}}{315 c^{4} \left (e x \right )^{\frac {11}{2}} \sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}\)	\(83\)
orering	\(-\frac {2 \left (-d x +c \right ) x \left (272 d^{4} x^{4}+136 c \,d^{3} x^{3}+102 d^{2} c^{2} x^{2}+85 c^{3} d x +35 c^{4}\right ) \sqrt {d x +c}}{315 c^{4} \left (e x \right )^{\frac {11}{2}} \sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}\)	\(83\)
default	\(-\frac {2 \sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}\, \left (272 d^{4} x^{4}+136 c \,d^{3} x^{3}+102 d^{2} c^{2} x^{2}+85 c^{3} d x +35 c^{4}\right )}{315 b \,x^{4} c^{4} \sqrt {d x +c}\, e^{5} \sqrt {e x}}\)	\(84\)

Input:

int((d*x+c)^(3/2)/(e*x)^(11/2)/(-b*d^2*x^2+b*c^2)^(1/2),x,method=_RETURNVE 
RBOSE)
 

Output:

-2/315*(-d*x+c)*x*(272*d^4*x^4+136*c*d^3*x^3+102*c^2*d^2*x^2+85*c^3*d*x+35 
*c^4)*(d*x+c)^(1/2)/c^4/(e*x)^(11/2)/(-b*d^2*x^2+b*c^2)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.40 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{11/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {2 \, {\left (272 \, d^{4} x^{4} + 136 \, c d^{3} x^{3} + 102 \, c^{2} d^{2} x^{2} + 85 \, c^{3} d x + 35 \, c^{4}\right )} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} \sqrt {e x}}{315 \, {\left (b c^{4} d e^{6} x^{6} + b c^{5} e^{6} x^{5}\right )}} \] Input:

integrate((d*x+c)^(3/2)/(e*x)^(11/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm 
="fricas")
 

Output:

-2/315*(272*d^4*x^4 + 136*c*d^3*x^3 + 102*c^2*d^2*x^2 + 85*c^3*d*x + 35*c^ 
4)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*sqrt(e*x)/(b*c^4*d*e^6*x^6 + b*c 
^5*e^6*x^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2}}{(e x)^{11/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(3/2)/(e*x)**(11/2)/(-b*d**2*x**2+b*c**2)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c+d x)^{3/2}}{(e x)^{11/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{\sqrt {-b d^{2} x^{2} + b c^{2}} \left (e x\right )^{\frac {11}{2}}} \,d x } \] Input:

integrate((d*x+c)^(3/2)/(e*x)^(11/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm 
="maxima")
 

Output:

integrate((d*x + c)^(3/2)/(sqrt(-b*d^2*x^2 + b*c^2)*(e*x)^(11/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (208) = 416\).

Time = 0.31 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.15 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{11/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {64 \, {\left (17 \, \sqrt {-b d e} b^{8} c^{5} d^{18} e^{17} {\left | d \right |} {\left | e \right |} - 153 \, \sqrt {-b d e} {\left (\sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {-b d e} - \sqrt {b c d^{2} e^{2} - {\left ({\left (d x + c\right )} d e - c d e\right )} b d e}\right )}^{2} b^{7} c^{4} d^{16} e^{15} {\left | d \right |} {\left | e \right |} + 612 \, \sqrt {-b d e} {\left (\sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {-b d e} - \sqrt {b c d^{2} e^{2} - {\left ({\left (d x + c\right )} d e - c d e\right )} b d e}\right )}^{4} b^{6} c^{3} d^{14} e^{13} {\left | d \right |} {\left | e \right |} - 1428 \, \sqrt {-b d e} {\left (\sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {-b d e} - \sqrt {b c d^{2} e^{2} - {\left ({\left (d x + c\right )} d e - c d e\right )} b d e}\right )}^{6} b^{5} c^{2} d^{12} e^{11} {\left | d \right |} {\left | e \right |} + 1827 \, \sqrt {-b d e} {\left (\sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {-b d e} - \sqrt {b c d^{2} e^{2} - {\left ({\left (d x + c\right )} d e - c d e\right )} b d e}\right )}^{8} b^{4} c d^{10} e^{9} {\left | d \right |} {\left | e \right |} - 315 \, \sqrt {-b d e} {\left (\sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {-b d e} - \sqrt {b c d^{2} e^{2} - {\left ({\left (d x + c\right )} d e - c d e\right )} b d e}\right )}^{10} b^{3} d^{8} e^{7} {\left | d \right |} {\left | e \right |}\right )} d^{4}}{315 \, {\left (b c d^{2} e^{2} - {\left (\sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {-b d e} - \sqrt {b c d^{2} e^{2} - {\left ({\left (d x + c\right )} d e - c d e\right )} b d e}\right )}^{2}\right )}^{9} e^{6} {\left | d \right |}} \] Input:

integrate((d*x+c)^(3/2)/(e*x)^(11/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm 
="giac")
 

Output:

-64/315*(17*sqrt(-b*d*e)*b^8*c^5*d^18*e^17*abs(d)*abs(e) - 153*sqrt(-b*d*e 
)*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-b*d*e) - sqrt(b*c*d^2*e^2 - ((d*x + c 
)*d*e - c*d*e)*b*d*e))^2*b^7*c^4*d^16*e^15*abs(d)*abs(e) + 612*sqrt(-b*d*e 
)*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-b*d*e) - sqrt(b*c*d^2*e^2 - ((d*x + c 
)*d*e - c*d*e)*b*d*e))^4*b^6*c^3*d^14*e^13*abs(d)*abs(e) - 1428*sqrt(-b*d* 
e)*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-b*d*e) - sqrt(b*c*d^2*e^2 - ((d*x + 
c)*d*e - c*d*e)*b*d*e))^6*b^5*c^2*d^12*e^11*abs(d)*abs(e) + 1827*sqrt(-b*d 
*e)*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-b*d*e) - sqrt(b*c*d^2*e^2 - ((d*x + 
 c)*d*e - c*d*e)*b*d*e))^8*b^4*c*d^10*e^9*abs(d)*abs(e) - 315*sqrt(-b*d*e) 
*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-b*d*e) - sqrt(b*c*d^2*e^2 - ((d*x + c) 
*d*e - c*d*e)*b*d*e))^10*b^3*d^8*e^7*abs(d)*abs(e))*d^4/((b*c*d^2*e^2 - (s 
qrt((d*x + c)*d*e - c*d*e)*sqrt(-b*d*e) - sqrt(b*c*d^2*e^2 - ((d*x + c)*d* 
e - c*d*e)*b*d*e))^2)^9*e^6*abs(d))
 

Mupad [B] (verification not implemented)

Time = 7.47 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.61 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{11/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {\sqrt {b\,c^2-b\,d^2\,x^2}\,\left (\frac {2\,\sqrt {c+d\,x}}{9\,b\,d\,e^5}+\frac {34\,x\,\sqrt {c+d\,x}}{63\,b\,c\,e^5}+\frac {68\,d\,x^2\,\sqrt {c+d\,x}}{105\,b\,c^2\,e^5}+\frac {272\,d^2\,x^3\,\sqrt {c+d\,x}}{315\,b\,c^3\,e^5}+\frac {544\,d^3\,x^4\,\sqrt {c+d\,x}}{315\,b\,c^4\,e^5}\right )}{x^5\,\sqrt {e\,x}+\frac {c\,x^4\,\sqrt {e\,x}}{d}} \] Input:

int((c + d*x)^(3/2)/((e*x)^(11/2)*(b*c^2 - b*d^2*x^2)^(1/2)),x)
 

Output:

-((b*c^2 - b*d^2*x^2)^(1/2)*((2*(c + d*x)^(1/2))/(9*b*d*e^5) + (34*x*(c + 
d*x)^(1/2))/(63*b*c*e^5) + (68*d*x^2*(c + d*x)^(1/2))/(105*b*c^2*e^5) + (2 
72*d^2*x^3*(c + d*x)^(1/2))/(315*b*c^3*e^5) + (544*d^3*x^4*(c + d*x)^(1/2) 
)/(315*b*c^4*e^5)))/(x^5*(e*x)^(1/2) + (c*x^4*(e*x)^(1/2))/d)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.46 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{11/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {2 \sqrt {e}\, \sqrt {b}\, \left (-35 \sqrt {x}\, \sqrt {-d x +c}\, c^{4}-85 \sqrt {x}\, \sqrt {-d x +c}\, c^{3} d x -102 \sqrt {x}\, \sqrt {-d x +c}\, c^{2} d^{2} x^{2}-136 \sqrt {x}\, \sqrt {-d x +c}\, c \,d^{3} x^{3}-272 \sqrt {x}\, \sqrt {-d x +c}\, d^{4} x^{4}+272 \sqrt {d}\, d^{4} i \,x^{5}\right )}{315 b \,c^{4} e^{6} x^{5}} \] Input:

int((d*x+c)^(3/2)/(e*x)^(11/2)/(-b*d^2*x^2+b*c^2)^(1/2),x)
 

Output:

(2*sqrt(e)*sqrt(b)*( - 35*sqrt(x)*sqrt(c - d*x)*c**4 - 85*sqrt(x)*sqrt(c - 
 d*x)*c**3*d*x - 102*sqrt(x)*sqrt(c - d*x)*c**2*d**2*x**2 - 136*sqrt(x)*sq 
rt(c - d*x)*c*d**3*x**3 - 272*sqrt(x)*sqrt(c - d*x)*d**4*x**4 + 272*sqrt(d 
)*d**4*i*x**5))/(315*b*c**4*e**6*x**5)