3.1.59 \(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^3} \, dx\)

Optimal. Leaf size=113 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} \sqrt {b c-a d}}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2} \]

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Rubi [A]  time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {378, 377, 208} \begin {gather*} \frac {3 a^2 \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} \sqrt {b c-a d}}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(3/2)/(c + d*x^2)^3,x]

[Out]

(x*(a + b*x^2)^(3/2))/(4*c*(c + d*x^2)^2) + (3*a*x*Sqrt[a + b*x^2])/(8*c^2*(c + d*x^2)) + (3*a^2*ArcTanh[(Sqrt
[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*Sqrt[b*c - a*d])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 378

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx &=\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {(3 a) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2} \, dx}{4 c}\\ &=\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {\left (3 a^2\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2}\\ &=\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 c^2}\\ &=\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} \sqrt {b c-a d}}\\ \end {align*}

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Mathematica [A]  time = 0.69, size = 163, normalized size = 1.44 \begin {gather*} \frac {x \sqrt {a+b x^2} \left (\frac {\sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \left (5 a c+3 a d x^2+2 b c x^2\right )}{\left (c+d x^2\right ) \sqrt {\frac {d x^2}{c}+1}}+\frac {3 a \sin ^{-1}\left (\frac {\sqrt {x^2 \left (\frac {d}{c}-\frac {b}{a}\right )}}{\sqrt {\frac {d x^2}{c}+1}}\right )}{\sqrt {\frac {x^2 (a d-b c)}{a c}}}\right )}{8 c^3 \sqrt {\frac {b x^2}{a}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^2)^(3/2)/(c + d*x^2)^3,x]

[Out]

(x*Sqrt[a + b*x^2]*((Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*(5*a*c + 2*b*c*x^2 + 3*a*d*x^2))/((c + d*x^2)*Sqrt[
1 + (d*x^2)/c]) + (3*a*ArcSin[Sqrt[(-(b/a) + d/c)*x^2]/Sqrt[1 + (d*x^2)/c]])/Sqrt[((-(b*c) + a*d)*x^2)/(a*c)])
)/(8*c^3*Sqrt[1 + (b*x^2)/a])

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IntegrateAlgebraic [B]  time = 3.19, size = 1323, normalized size = 11.71 \begin {gather*} -\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} d x^2}{\sqrt {c} \sqrt {a d-b c}}-\frac {d \sqrt {b x^2+a} x}{\sqrt {c} \sqrt {a d-b c}}+\frac {\sqrt {b} \sqrt {c}}{\sqrt {a d-b c}}\right ) a^2}{8 c^{5/2} \sqrt {a d-b c}}+\frac {3 a^2}{8 c^2 d x \left (\sqrt {b x^2+a}-\sqrt {b} x\right )}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt {b} d x^2}{\sqrt {c} \sqrt {a d-b c}}-\frac {d \sqrt {b x^2+a} x}{\sqrt {c} \sqrt {a d-b c}}+\frac {\sqrt {b} \sqrt {c}}{\sqrt {a d-b c}}\right ) a}{c^{3/2} d \sqrt {a d-b c}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {b} d x^2}{\sqrt {c} \sqrt {b c-a d}}-\frac {d \sqrt {b x^2+a} x}{\sqrt {c} \sqrt {b c-a d}}+\frac {\sqrt {b} \sqrt {c}}{\sqrt {b c-a d}}\right ) a}{c^{3/2} d \sqrt {b c-a d}}+\frac {-\frac {a^4}{8 c d^2}-\frac {11 b x^2 a^3}{8 c d^2}+\frac {5 \sqrt {b} x \sqrt {b x^2+a} a^3}{8 c d^2}-\frac {3 b^2 x^4 a^2}{2 c d^2}+\frac {3 b^{3/2} x^3 \sqrt {b x^2+a} a^2}{2 c d^2}}{x^3 \left (\sqrt {b x^2+a}-\sqrt {b} x\right ) \left (8 b^2 x^4-8 b^{3/2} \sqrt {b x^2+a} x^3+8 a b x^2-4 a \sqrt {b} \sqrt {b x^2+a} x+a^2\right )}+\frac {-\frac {4 b^4 x^8}{d}+\frac {4 b^{7/2} \sqrt {b x^2+a} x^7}{d}-\frac {11 a b^3 x^6}{d}+\frac {9 a b^{5/2} \sqrt {b x^2+a} x^5}{d}-\frac {41 a^2 b^2 x^4}{4 d}+\frac {25 a^2 b^{3/2} \sqrt {b x^2+a} x^3}{4 d}-\frac {7 a^3 b x^2}{2 d}+\frac {5 a^3 \sqrt {b} \sqrt {b x^2+a} x}{4 d}-\frac {a^4}{4 d}}{x \left (d x^2+c\right )^2 \left (\sqrt {b x^2+a}-\sqrt {b} x\right ) \left (2 b x^2-2 \sqrt {b} \sqrt {b x^2+a} x+a\right )^2}+\frac {-\frac {16 b^5 x^{10}}{d^2}+\frac {16 b^{9/2} \sqrt {b x^2+a} x^9}{d^2}-\frac {28 a b^4 x^8}{d^2}+\frac {20 a b^{7/2} \sqrt {b x^2+a} x^7}{d^2}-\frac {8 a^2 b^3 x^6}{d^2}+\frac {27 a^3 b^2 x^4}{4 d^2}-\frac {21 a^3 b^{3/2} \sqrt {b x^2+a} x^3}{4 d^2}+\frac {23 a^4 b x^2}{8 d^2}-\frac {7 a^4 \sqrt {b} \sqrt {b x^2+a} x}{8 d^2}+\frac {a^5}{8 d^2}}{x^3 \left (d x^2+c\right ) \left (\sqrt {b x^2+a}-\sqrt {b} x\right ) \left (2 b x^2-2 \sqrt {b} \sqrt {b x^2+a} x+a\right ) \left (8 b^2 x^4-8 b^{3/2} \sqrt {b x^2+a} x^3+8 a b x^2-4 a \sqrt {b} \sqrt {b x^2+a} x+a^2\right )}-\frac {4 b^2 \tan ^{-1}\left (\frac {\sqrt {b} d x^2}{\sqrt {c} \sqrt {a d-b c}}-\frac {d \sqrt {b x^2+a} x}{\sqrt {c} \sqrt {a d-b c}}+\frac {\sqrt {b} \sqrt {c}}{\sqrt {a d-b c}}\right )}{\sqrt {c} d^2 \sqrt {a d-b c}}-\frac {4 b^2 \tanh ^{-1}\left (\frac {\sqrt {b} d x^2}{\sqrt {c} \sqrt {b c-a d}}-\frac {d \sqrt {b x^2+a} x}{\sqrt {c} \sqrt {b c-a d}}+\frac {\sqrt {b} \sqrt {c}}{\sqrt {b c-a d}}\right )}{\sqrt {c} d^2 \sqrt {b c-a d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x^2)^(3/2)/(c + d*x^2)^3,x]

[Out]

(3*a^2)/(8*c^2*d*x*(-(Sqrt[b]*x) + Sqrt[a + b*x^2])) + (-1/8*a^4/(c*d^2) - (11*a^3*b*x^2)/(8*c*d^2) - (3*a^2*b
^2*x^4)/(2*c*d^2) + (5*a^3*Sqrt[b]*x*Sqrt[a + b*x^2])/(8*c*d^2) + (3*a^2*b^(3/2)*x^3*Sqrt[a + b*x^2])/(2*c*d^2
))/(x^3*(-(Sqrt[b]*x) + Sqrt[a + b*x^2])*(a^2 + 8*a*b*x^2 + 8*b^2*x^4 - 4*a*Sqrt[b]*x*Sqrt[a + b*x^2] - 8*b^(3
/2)*x^3*Sqrt[a + b*x^2])) + (-1/4*a^4/d - (7*a^3*b*x^2)/(2*d) - (41*a^2*b^2*x^4)/(4*d) - (11*a*b^3*x^6)/d - (4
*b^4*x^8)/d + (5*a^3*Sqrt[b]*x*Sqrt[a + b*x^2])/(4*d) + (25*a^2*b^(3/2)*x^3*Sqrt[a + b*x^2])/(4*d) + (9*a*b^(5
/2)*x^5*Sqrt[a + b*x^2])/d + (4*b^(7/2)*x^7*Sqrt[a + b*x^2])/d)/(x*(c + d*x^2)^2*(-(Sqrt[b]*x) + Sqrt[a + b*x^
2])*(a + 2*b*x^2 - 2*Sqrt[b]*x*Sqrt[a + b*x^2])^2) + (a^5/(8*d^2) + (23*a^4*b*x^2)/(8*d^2) + (27*a^3*b^2*x^4)/
(4*d^2) - (8*a^2*b^3*x^6)/d^2 - (28*a*b^4*x^8)/d^2 - (16*b^5*x^10)/d^2 - (7*a^4*Sqrt[b]*x*Sqrt[a + b*x^2])/(8*
d^2) - (21*a^3*b^(3/2)*x^3*Sqrt[a + b*x^2])/(4*d^2) + (20*a*b^(7/2)*x^7*Sqrt[a + b*x^2])/d^2 + (16*b^(9/2)*x^9
*Sqrt[a + b*x^2])/d^2)/(x^3*(c + d*x^2)*(-(Sqrt[b]*x) + Sqrt[a + b*x^2])*(a + 2*b*x^2 - 2*Sqrt[b]*x*Sqrt[a + b
*x^2])*(a^2 + 8*a*b*x^2 + 8*b^2*x^4 - 4*a*Sqrt[b]*x*Sqrt[a + b*x^2] - 8*b^(3/2)*x^3*Sqrt[a + b*x^2])) - (3*a^2
*ArcTan[(Sqrt[b]*Sqrt[c])/Sqrt[-(b*c) + a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[-(b*c) + a*d]) - (d*x*Sqrt[a + b*
x^2])/(Sqrt[c]*Sqrt[-(b*c) + a*d])])/(8*c^(5/2)*Sqrt[-(b*c) + a*d]) - (4*b^2*ArcTan[(Sqrt[b]*Sqrt[c])/Sqrt[-(b
*c) + a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[-(b*c) + a*d]) - (d*x*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[-(b*c) + a*d])
])/(Sqrt[c]*d^2*Sqrt[-(b*c) + a*d]) + (3*a*b*ArcTan[(Sqrt[b]*Sqrt[c])/Sqrt[-(b*c) + a*d] + (Sqrt[b]*d*x^2)/(Sq
rt[c]*Sqrt[-(b*c) + a*d]) - (d*x*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[-(b*c) + a*d])])/(c^(3/2)*d*Sqrt[-(b*c) + a*d]
) - (4*b^2*ArcTanh[(Sqrt[b]*Sqrt[c])/Sqrt[b*c - a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[b*c - a*d]) - (d*x*Sqrt[a
+ b*x^2])/(Sqrt[c]*Sqrt[b*c - a*d])])/(Sqrt[c]*d^2*Sqrt[b*c - a*d]) + (3*a*b*ArcTanh[(Sqrt[b]*Sqrt[c])/Sqrt[b
*c - a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[b*c - a*d]) - (d*x*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[b*c - a*d])])/(c^(
3/2)*d*Sqrt[b*c - a*d])

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fricas [B]  time = 1.21, size = 526, normalized size = 4.65 \begin {gather*} \left [\frac {3 \, {\left (a^{2} d^{2} x^{4} + 2 \, a^{2} c d x^{2} + a^{2} c^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3} + 5 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a}}{32 \, {\left (b c^{6} - a c^{5} d + {\left (b c^{4} d^{2} - a c^{3} d^{3}\right )} x^{4} + 2 \, {\left (b c^{5} d - a c^{4} d^{2}\right )} x^{2}\right )}}, -\frac {3 \, {\left (a^{2} d^{2} x^{4} + 2 \, a^{2} c d x^{2} + a^{2} c^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3} + 5 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b c^{6} - a c^{5} d + {\left (b c^{4} d^{2} - a c^{3} d^{3}\right )} x^{4} + 2 \, {\left (b c^{5} d - a c^{4} d^{2}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

[1/32*(3*(a^2*d^2*x^4 + 2*a^2*c*d*x^2 + a^2*c^2)*sqrt(b*c^2 - a*c*d)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^
4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*((2*b*c - a*d)*x^3 + a*c*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a
))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 4*((2*b^2*c^3 + a*b*c^2*d - 3*a^2*c*d^2)*x^3 + 5*(a*b*c^3 - a^2*c^2*d)*x)*sq
rt(b*x^2 + a))/(b*c^6 - a*c^5*d + (b*c^4*d^2 - a*c^3*d^3)*x^4 + 2*(b*c^5*d - a*c^4*d^2)*x^2), -1/16*(3*(a^2*d^
2*x^4 + 2*a^2*c*d*x^2 + a^2*c^2)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 + a*c
)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 2*((2*b^2*c^3 + a*b*c^2*d - 3*a^2*c*d^2
)*x^3 + 5*(a*b*c^3 - a^2*c^2*d)*x)*sqrt(b*x^2 + a))/(b*c^6 - a*c^5*d + (b*c^4*d^2 - a*c^3*d^3)*x^4 + 2*(b*c^5*
d - a*c^4*d^2)*x^2)]

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giac [B]  time = 3.72, size = 451, normalized size = 3.99 \begin {gather*} -\frac {3 \, a^{2} \sqrt {b} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{8 \, \sqrt {-b^{2} c^{2} + a b c d} c^{2}} + \frac {8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {5}{2}} c^{2} d - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} \sqrt {b} d^{3} + 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{3} + 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{2} d - 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c d^{2} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} \sqrt {b} d^{3} + 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{2} d + 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c d^{2} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} \sqrt {b} d^{3} + 2 \, a^{4} b^{\frac {3}{2}} c d^{2} + 3 \, a^{5} \sqrt {b} d^{3}}{4 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2} c^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

-3/8*a^2*sqrt(b)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/(sqrt(
-b^2*c^2 + a*b*c*d)*c^2) + 1/4*(8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c^2*d - 3*(sqrt(b)*x - sqrt(b*x^2 +
a))^6*a^2*sqrt(b)*d^3 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^3 + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b
^(5/2)*c^2*d - 18*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*d^2 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*sq
rt(b)*d^3 + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c^2*d + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(3/
2)*c*d^2 - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*sqrt(b)*d^3 + 2*a^4*b^(3/2)*c*d^2 + 3*a^5*sqrt(b)*d^3)/(((sqr
t(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d
+ a^2*d)^2*c^2*d^2)

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maple [B]  time = 0.02, size = 9059, normalized size = 80.17 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(3/2)/(d*x^2+c)^3,x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(3/2)/(d*x^2 + c)^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(3/2)/(c + d*x^2)^3,x)

[Out]

int((a + b*x^2)^(3/2)/(c + d*x^2)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{\left (c + d x^{2}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(3/2)/(d*x**2+c)**3,x)

[Out]

Integral((a + b*x**2)**(3/2)/(c + d*x**2)**3, x)

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