Integrand size = 26, antiderivative size = 320 \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^2 e^2 (e x)^{3/2}}{d \left (b c^2+a d^2\right ) \sqrt {c+d x}}+\frac {\left (3 b c^2+a d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x}}{b d^2 \left (b c^2+a d^2\right )}-\frac {(-a)^{5/4} e^{7/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{b^{3/2} \left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2}}-\frac {3 c e^{7/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{b d^{5/2}}-\frac {(-a)^{5/4} e^{7/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{b^{3/2} \left (\sqrt {b} c+\sqrt {-a} d\right )^{3/2}} \] Output:
-2*c^2*e^2*(e*x)^(3/2)/d/(a*d^2+b*c^2)/(d*x+c)^(1/2)+(a*d^2+3*b*c^2)*e^3*( e*x)^(1/2)*(d*x+c)^(1/2)/b/d^2/(a*d^2+b*c^2)-(-a)^(5/4)*e^(7/2)*arctan((b^ (1/2)*c-(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/ b^(3/2)/(b^(1/2)*c-(-a)^(1/2)*d)^(3/2)-3*c*e^(7/2)*arctanh(d^(1/2)*(e*x)^( 1/2)/e^(1/2)/(d*x+c)^(1/2))/b/d^(5/2)-(-a)^(5/4)*e^(7/2)*arctanh((b^(1/2)* c+(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/b^(3/2 )/(b^(1/2)*c+(-a)^(1/2)*d)^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.80 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.33 \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {e^3 \sqrt {e x} \left (4 \sqrt {d} \sqrt {x} \left (a d^2 (c+d x)+b c^2 (3 c+d x)\right )+24 c \left (b c^2+a d^2\right ) \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}-\sqrt {c+d x}}\right )+a^2 c d^{5/2} \sqrt {c+d x} \text {RootSum}\left [a d^4-4 a d^3 \text {$\#$1}^2+16 b c^2 \text {$\#$1}^4+6 a d^2 \text {$\#$1}^4-4 a d \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {d^3 \log (x)-2 d^3 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right )-7 d^2 \log (x) \text {$\#$1}^2+14 d^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+7 d \log (x) \text {$\#$1}^4-14 d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-\log (x) \text {$\#$1}^6+2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^6}{a d^3 \text {$\#$1}-8 b c^2 \text {$\#$1}^3-3 a d^2 \text {$\#$1}^3+3 a d \text {$\#$1}^5-a \text {$\#$1}^7}\&\right ]\right )}{4 b d^{5/2} \left (b c^2+a d^2\right ) \sqrt {x} \sqrt {c+d x}} \] Input:
Integrate[(e*x)^(7/2)/((c + d*x)^(3/2)*(a + b*x^2)),x]
Output:
(e^3*Sqrt[e*x]*(4*Sqrt[d]*Sqrt[x]*(a*d^2*(c + d*x) + b*c^2*(3*c + d*x)) + 24*c*(b*c^2 + a*d^2)*Sqrt[c + d*x]*ArcTanh[(Sqrt[d]*Sqrt[x])/(Sqrt[c] - Sq rt[c + d*x])] + a^2*c*d^(5/2)*Sqrt[c + d*x]*RootSum[a*d^4 - 4*a*d^3*#1^2 + 16*b*c^2*#1^4 + 6*a*d^2*#1^4 - 4*a*d*#1^6 + a*#1^8 & , (d^3*Log[x] - 2*d^ 3*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1] - 7*d^2*Log[x]*#1^2 + 14*d^2* Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 + 7*d*Log[x]*#1^4 - 14*d*L og[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^4 - Log[x]*#1^6 + 2*Log[-Sqrt [c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^6)/(a*d^3*#1 - 8*b*c^2*#1^3 - 3*a*d^2 *#1^3 + 3*a*d*#1^5 - a*#1^7) & ]))/(4*b*d^(5/2)*(b*c^2 + a*d^2)*Sqrt[x]*Sq rt[c + d*x])
Leaf count is larger than twice the leaf count of optimal. \(768\) vs. \(2(320)=640\).
Time = 2.49 (sec) , antiderivative size = 768, normalized size of antiderivative = 2.40, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {612, 57, 60, 60, 65, 221, 2353, 2009}
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.
| \(\displaystyle \int \frac {(e x)^{7/2}}{\left (a+b x^2\right ) (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 612 |
| \(\displaystyle \frac {e \int \frac {(e x)^{5/2} (a d+b c x)}{\sqrt {c+d x} \left (b x^2+a\right )}dx}{a d^2+b c^2}-\frac {c d e \int \frac {(e x)^{5/2}}{(c+d x)^{3/2}}dx}{a d^2+b c^2}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {e \int \frac {(e x)^{5/2} (a d+b c x)}{\sqrt {c+d x} \left (b x^2+a\right )}dx}{a d^2+b c^2}-\frac {c d e \left (\frac {5 e \int \frac {(e x)^{3/2}}{\sqrt {c+d x}}dx}{d}-\frac {2 (e x)^{5/2}}{d \sqrt {c+d x}}\right )}{a d^2+b c^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {e \int \frac {(e x)^{5/2} (a d+b c x)}{\sqrt {c+d x} \left (b x^2+a\right )}dx}{a d^2+b c^2}-\frac {c d e \left (\frac {5 e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx}{4 d}\right )}{d}-\frac {2 (e x)^{5/2}}{d \sqrt {c+d x}}\right )}{a d^2+b c^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {e \int \frac {(e x)^{5/2} (a d+b c x)}{\sqrt {c+d x} \left (b x^2+a\right )}dx}{a d^2+b c^2}-\frac {c d e \left (\frac {5 e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{d}-\frac {2 (e x)^{5/2}}{d \sqrt {c+d x}}\right )}{a d^2+b c^2}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \frac {e \int \frac {(e x)^{5/2} (a d+b c x)}{\sqrt {c+d x} \left (b x^2+a\right )}dx}{a d^2+b c^2}-\frac {c d e \left (\frac {5 e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{d}-\frac {2 (e x)^{5/2}}{d \sqrt {c+d x}}\right )}{a d^2+b c^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {e \int \frac {(e x)^{5/2} (a d+b c x)}{\sqrt {c+d x} \left (b x^2+a\right )}dx}{a d^2+b c^2}-\frac {c d e \left (\frac {5 e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )}{4 d}\right )}{d}-\frac {2 (e x)^{5/2}}{d \sqrt {c+d x}}\right )}{a d^2+b c^2}\) |
| \(\Big \downarrow \) 2353 |
| \(\displaystyle \frac {e \int \left (\frac {\left (\sqrt {-a} a d-a \sqrt {b} c\right ) (e x)^{5/2}}{2 a \left (\sqrt {-a}-\sqrt {b} x\right ) \sqrt {c+d x}}+\frac {\left (a \sqrt {b} c+\sqrt {-a} a d\right ) (e x)^{5/2}}{2 a \left (\sqrt {b} x+\sqrt {-a}\right ) \sqrt {c+d x}}\right )dx}{a d^2+b c^2}-\frac {c d e \left (\frac {5 e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )}{4 d}\right )}{d}-\frac {2 (e x)^{5/2}}{d \sqrt {c+d x}}\right )}{a d^2+b c^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e \left (-\frac {(-a)^{5/4} e^{5/2} \left (\sqrt {-a} d+\sqrt {b} c\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {\sqrt {b} c-\sqrt {-a} d}}+\frac {e^{5/2} \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (-4 \sqrt {-a} \sqrt {b} c d-8 a d^2+3 b c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{5/2}}+\frac {e^{5/2} \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (4 \sqrt {-a} \sqrt {b} c d-8 a d^2+3 b c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{5/2}}-\frac {(-a)^{5/4} e^{5/2} \left (\sqrt {b} c-\sqrt {-a} d\right ) \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {\sqrt {-a} d+\sqrt {b} c}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x} \left (-7 \sqrt {-a} \sqrt {b} c d-4 a d^2+3 b c^2\right )}{8 b d^2}-\frac {e^2 \sqrt {e x} \sqrt {c+d x} \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (4 \sqrt {-a} d+3 \sqrt {b} c\right )}{8 b d^2}-\frac {1}{4} e (e x)^{3/2} \sqrt {c+d x} \left (\frac {\sqrt {-a}}{\sqrt {b}}-\frac {c}{d}\right )+\frac {1}{4} e (e x)^{3/2} \sqrt {c+d x} \left (\frac {\sqrt {-a}}{\sqrt {b}}+\frac {c}{d}\right )\right )}{a d^2+b c^2}-\frac {c d e \left (\frac {5 e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )}{4 d}\right )}{d}-\frac {2 (e x)^{5/2}}{d \sqrt {c+d x}}\right )}{a d^2+b c^2}\) |
Input:
Int[(e*x)^(7/2)/((c + d*x)^(3/2)*(a + b*x^2)),x]
Output:
-((c*d*e*((-2*(e*x)^(5/2))/(d*Sqrt[c + d*x]) + (5*e*(((e*x)^(3/2)*Sqrt[c + d*x])/(2*d) - (3*c*e*((Sqrt[e*x]*Sqrt[c + d*x])/d - (c*Sqrt[e]*ArcTanh[(S qrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/d^(3/2)))/(4*d)))/d))/(b*c^2 + a*d^2)) + (e*(-1/8*((Sqrt[b]*c + Sqrt[-a]*d)*(3*Sqrt[b]*c + 4*Sqrt[-a]*d) *e^2*Sqrt[e*x]*Sqrt[c + d*x])/(b*d^2) - ((3*b*c^2 - 7*Sqrt[-a]*Sqrt[b]*c*d - 4*a*d^2)*e^2*Sqrt[e*x]*Sqrt[c + d*x])/(8*b*d^2) - ((Sqrt[-a]/Sqrt[b] - c/d)*e*(e*x)^(3/2)*Sqrt[c + d*x])/4 + ((Sqrt[-a]/Sqrt[b] + c/d)*e*(e*x)^(3 /2)*Sqrt[c + d*x])/4 - ((-a)^(5/4)*(Sqrt[b]*c + Sqrt[-a]*d)*e^(5/2)*ArcTan [(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x ])])/(b^(3/2)*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]) + ((Sqrt[b]*c - Sqrt[-a]*d)*(3 *b*c^2 - 4*Sqrt[-a]*Sqrt[b]*c*d - 8*a*d^2)*e^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e *x])/(Sqrt[e]*Sqrt[c + d*x])])/(8*b^(3/2)*d^(5/2)) + ((Sqrt[b]*c + Sqrt[-a ]*d)*(3*b*c^2 + 4*Sqrt[-a]*Sqrt[b]*c*d - 8*a*d^2)*e^(5/2)*ArcTanh[(Sqrt[d] *Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(8*b^(3/2)*d^(5/2)) - ((-a)^(5/4)*(S qrt[b]*c - Sqrt[-a]*d)*e^(5/2)*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[ e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(b^(3/2)*Sqrt[Sqrt[b]*c + Sqrt[ -a]*d])))/(b*c^2 + a*d^2)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S ymbol] :> Simp[(-e)*c*(d/(b*c^2 + a*d^2)) Int[(e*x)^(m - 1)*(c + d*x)^n, x], x] + Simp[e/(b*c^2 + a*d^2) Int[(e*x)^(m - 1)*(c + d*x)^(n + 1)*((a*d + b*c*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[n, -1] & & GtQ[m, 0] && !IntegerQ[m] && !IntegerQ[n]
Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) ^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs. \(2(248)=496\).
Time = 0.52 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.83
| method | result | size |
| risch | \(\frac {x \sqrt {d x +c}\, e^{4}}{d^{2} b \sqrt {e x}}+\frac {\left (-\frac {3 c \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{2 b \,d^{2} \sqrt {d e}}-\frac {2 b \,c^{3} \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{d^{3} \left (\sqrt {-a b}\, d +b c \right ) \left (\sqrt {-a b}\, d -b c \right ) e \left (x +\frac {c}{d}\right )}-\frac {a^{2} \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 b \left (\sqrt {-a b}\, d +b c \right ) \sqrt {-a b}\, \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}-\frac {a^{2} \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 b \left (\sqrt {-a b}\, d -b c \right ) \sqrt {-a b}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\right ) e^{4} \sqrt {\left (d x +c \right ) e x}}{\sqrt {e x}\, \sqrt {d x +c}}\) | \(587\) |
| default | \(\text {Expression too large to display}\) | \(2829\) |
Input:
int((e*x)^(7/2)/(d*x+c)^(3/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/d^2/b*x*(d*x+c)^(1/2)*e^4/(e*x)^(1/2)+(-3/2/b/d^2*c*ln((1/2*c*e+d*e*x)/( d*e)^(1/2)+(d*e*x^2+c*e*x)^(1/2))/(d*e)^(1/2)-2*b/d^3*c^3/((-a*b)^(1/2)*d+ b*c)/((-a*b)^(1/2)*d-b*c)/e/(x+c/d)*(d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2)-1/2/ b*a^2/((-a*b)^(1/2)*d+b*c)/(-a*b)^(1/2)/(-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)* ln((-2*e*(a*d-c*(-a*b)^(1/2))/b+e*(2*(-a*b)^(1/2)*d+b*c)/b*(x-(-a*b)^(1/2) /b)+2*(-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)*(d*e*(x-(-a*b)^(1/2)/b)^2+e*(2*(-a *b)^(1/2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2))/(x- (-a*b)^(1/2)/b))-1/2/b*a^2/((-a*b)^(1/2)*d-b*c)/(-a*b)^(1/2)/(-e*(a*d+c*(- a*b)^(1/2))/b)^(1/2)*ln((-2*e*(a*d+c*(-a*b)^(1/2))/b+e*(-2*(-a*b)^(1/2)*d+ b*c)/b*(x+(-a*b)^(1/2)/b)+2*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*(d*e*(x+(-a* b)^(1/2)/b)^2+e*(-2*(-a*b)^(1/2)*d+b*c)/b*(x+(-a*b)^(1/2)/b)-e*(a*d+c*(-a* b)^(1/2))/b)^(1/2))/(x+(-a*b)^(1/2)/b)))*e^4*((d*x+c)*e*x)^(1/2)/(e*x)^(1/ 2)/(d*x+c)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 3313 vs. \(2 (248) = 496\).
Time = 2.39 (sec) , antiderivative size = 6625, normalized size of antiderivative = 20.70 \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(7/2)/(d*x+c)^(3/2)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {\left (e x\right )^{\frac {7}{2}}}{\left (a + b x^{2}\right ) \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x)**(7/2)/(d*x+c)**(3/2)/(b*x**2+a),x)
Output:
Integral((e*x)**(7/2)/((a + b*x**2)*(c + d*x)**(3/2)), x)
\[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x)^(7/2)/(d*x+c)^(3/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((e*x)^(7/2)/((b*x^2 + a)*(d*x + c)^(3/2)), x)
Exception generated. \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x)^(7/2)/(d*x+c)^(3/2)/(b*x^2+a),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}}{\left (b\,x^2+a\right )\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int((e*x)^(7/2)/((a + b*x^2)*(c + d*x)^(3/2)),x)
Output:
int((e*x)^(7/2)/((a + b*x^2)*(c + d*x)^(3/2)), x)
\[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, x^{3}}{\sqrt {d x +c}\, a c +\sqrt {d x +c}\, a d x +\sqrt {d x +c}\, b c \,x^{2}+\sqrt {d x +c}\, b d \,x^{3}}d x \right ) e^{3} \] Input:
int((e*x)^(7/2)/(d*x+c)^(3/2)/(b*x^2+a),x)
Output:
sqrt(e)*int((sqrt(x)*x**3)/(sqrt(c + d*x)*a*c + sqrt(c + d*x)*a*d*x + sqrt (c + d*x)*b*c*x**2 + sqrt(c + d*x)*b*d*x**3),x)*e**3