3.1.63 \(\int (a+b x^2)^{5/2} (c+d x^2)^2 \, dx\)

Optimal. Leaf size=241 \[ \frac {x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{480 b^2}+\frac {a x \left (a+b x^2\right )^{3/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{384 b^2}+\frac {a^2 x \sqrt {a+b x^2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^2}+\frac {a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}+\frac {3 d x \left (a+b x^2\right )^{7/2} (4 b c-a d)}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b} \]

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Rubi [A]  time = 0.15, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {416, 388, 195, 217, 206} \begin {gather*} \frac {x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{480 b^2}+\frac {a x \left (a+b x^2\right )^{3/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{384 b^2}+\frac {a^2 x \sqrt {a+b x^2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^2}+\frac {a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}+\frac {3 d x \left (a+b x^2\right )^{7/2} (4 b c-a d)}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)*(c + d*x^2)^2,x]

[Out]

(a^2*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*Sqrt[a + b*x^2])/(256*b^2) + (a*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*
d^2)*x*(a + b*x^2)^(3/2))/(384*b^2) + ((80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(5/2))/(480*b^2) +
(3*d*(4*b*c - a*d)*x*(a + b*x^2)^(7/2))/(80*b^2) + (d*x*(a + b*x^2)^(7/2)*(c + d*x^2))/(10*b) + (a^3*(80*b^2*c
^2 - 20*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(5/2))

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 388

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

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rubi steps

\begin {align*} \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx &=\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\int \left (a+b x^2\right )^{5/2} \left (c (10 b c-a d)+3 d (4 b c-a d) x^2\right ) \, dx}{10 b}\\ &=\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}-\frac {(3 a d (4 b c-a d)-8 b c (10 b c-a d)) \int \left (a+b x^2\right )^{5/2} \, dx}{80 b^2}\\ &=\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{96 b^2}\\ &=\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \sqrt {a+b x^2} \, dx}{128 b^2}\\ &=\frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{256 b^2}\\ &=\frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{256 b^2}\\ &=\frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 2.80, size = 158, normalized size = 0.66 \begin {gather*} \frac {a x \sqrt {a+b x^2} \left (10 b x^2 \left (c+d x^2\right )^2 \, _3F_2\left (-\frac {3}{2},\frac {3}{2},2;1,\frac {9}{2};-\frac {b x^2}{a}\right )+20 b x^2 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \, _2F_1\left (-\frac {3}{2},\frac {3}{2};\frac {9}{2};-\frac {b x^2}{a}\right )+7 a \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \, _2F_1\left (-\frac {5}{2},\frac {1}{2};\frac {7}{2};-\frac {b x^2}{a}\right )\right )}{105 \sqrt {\frac {b x^2}{a}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^2)^(5/2)*(c + d*x^2)^2,x]

[Out]

(a*x*Sqrt[a + b*x^2]*(7*a*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*Hypergeometric2F1[-5/2, 1/2, 7/2, -((b*x^2)/a)] +
20*b*x^2*(2*c^2 + 3*c*d*x^2 + d^2*x^4)*Hypergeometric2F1[-3/2, 3/2, 9/2, -((b*x^2)/a)] + 10*b*x^2*(c + d*x^2)^
2*HypergeometricPFQ[{-3/2, 3/2, 2}, {1, 9/2}, -((b*x^2)/a)]))/(105*Sqrt[1 + (b*x^2)/a])

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IntegrateAlgebraic [A]  time = 0.38, size = 214, normalized size = 0.89 \begin {gather*} \frac {\left (-3 a^5 d^2+20 a^4 b c d-80 a^3 b^2 c^2\right ) \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{256 b^{5/2}}+\frac {\sqrt {a+b x^2} \left (-45 a^4 d^2 x+300 a^3 b c d x+30 a^3 b d^2 x^3+2640 a^2 b^2 c^2 x+2360 a^2 b^2 c d x^3+744 a^2 b^2 d^2 x^5+2080 a b^3 c^2 x^3+2720 a b^3 c d x^5+1008 a b^3 d^2 x^7+640 b^4 c^2 x^5+960 b^4 c d x^7+384 b^4 d^2 x^9\right )}{3840 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x^2)^(5/2)*(c + d*x^2)^2,x]

[Out]

(Sqrt[a + b*x^2]*(2640*a^2*b^2*c^2*x + 300*a^3*b*c*d*x - 45*a^4*d^2*x + 2080*a*b^3*c^2*x^3 + 2360*a^2*b^2*c*d*
x^3 + 30*a^3*b*d^2*x^3 + 640*b^4*c^2*x^5 + 2720*a*b^3*c*d*x^5 + 744*a^2*b^2*d^2*x^5 + 960*b^4*c*d*x^7 + 1008*a
*b^3*d^2*x^7 + 384*b^4*d^2*x^9))/(3840*b^2) + ((-80*a^3*b^2*c^2 + 20*a^4*b*c*d - 3*a^5*d^2)*Log[-(Sqrt[b]*x) +
Sqrt[a + b*x^2]])/(256*b^(5/2))

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fricas [A]  time = 1.79, size = 420, normalized size = 1.74 \begin {gather*} \left [\frac {15 \, {\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (384 \, b^{5} d^{2} x^{9} + 48 \, {\left (20 \, b^{5} c d + 21 \, a b^{4} d^{2}\right )} x^{7} + 8 \, {\left (80 \, b^{5} c^{2} + 340 \, a b^{4} c d + 93 \, a^{2} b^{3} d^{2}\right )} x^{5} + 10 \, {\left (208 \, a b^{4} c^{2} + 236 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{3} + 15 \, {\left (176 \, a^{2} b^{3} c^{2} + 20 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{7680 \, b^{3}}, -\frac {15 \, {\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (384 \, b^{5} d^{2} x^{9} + 48 \, {\left (20 \, b^{5} c d + 21 \, a b^{4} d^{2}\right )} x^{7} + 8 \, {\left (80 \, b^{5} c^{2} + 340 \, a b^{4} c d + 93 \, a^{2} b^{3} d^{2}\right )} x^{5} + 10 \, {\left (208 \, a b^{4} c^{2} + 236 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{3} + 15 \, {\left (176 \, a^{2} b^{3} c^{2} + 20 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{3840 \, b^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c)^2,x, algorithm="fricas")

[Out]

[1/7680*(15*(80*a^3*b^2*c^2 - 20*a^4*b*c*d + 3*a^5*d^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a
) + 2*(384*b^5*d^2*x^9 + 48*(20*b^5*c*d + 21*a*b^4*d^2)*x^7 + 8*(80*b^5*c^2 + 340*a*b^4*c*d + 93*a^2*b^3*d^2)*
x^5 + 10*(208*a*b^4*c^2 + 236*a^2*b^3*c*d + 3*a^3*b^2*d^2)*x^3 + 15*(176*a^2*b^3*c^2 + 20*a^3*b^2*c*d - 3*a^4*
b*d^2)*x)*sqrt(b*x^2 + a))/b^3, -1/3840*(15*(80*a^3*b^2*c^2 - 20*a^4*b*c*d + 3*a^5*d^2)*sqrt(-b)*arctan(sqrt(-
b)*x/sqrt(b*x^2 + a)) - (384*b^5*d^2*x^9 + 48*(20*b^5*c*d + 21*a*b^4*d^2)*x^7 + 8*(80*b^5*c^2 + 340*a*b^4*c*d
+ 93*a^2*b^3*d^2)*x^5 + 10*(208*a*b^4*c^2 + 236*a^2*b^3*c*d + 3*a^3*b^2*d^2)*x^3 + 15*(176*a^2*b^3*c^2 + 20*a^
3*b^2*c*d - 3*a^4*b*d^2)*x)*sqrt(b*x^2 + a))/b^3]

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giac [A]  time = 0.66, size = 221, normalized size = 0.92 \begin {gather*} \frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, b^{2} d^{2} x^{2} + \frac {20 \, b^{10} c d + 21 \, a b^{9} d^{2}}{b^{8}}\right )} x^{2} + \frac {80 \, b^{10} c^{2} + 340 \, a b^{9} c d + 93 \, a^{2} b^{8} d^{2}}{b^{8}}\right )} x^{2} + \frac {5 \, {\left (208 \, a b^{9} c^{2} + 236 \, a^{2} b^{8} c d + 3 \, a^{3} b^{7} d^{2}\right )}}{b^{8}}\right )} x^{2} + \frac {15 \, {\left (176 \, a^{2} b^{8} c^{2} + 20 \, a^{3} b^{7} c d - 3 \, a^{4} b^{6} d^{2}\right )}}{b^{8}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c)^2,x, algorithm="giac")

[Out]

1/3840*(2*(4*(6*(8*b^2*d^2*x^2 + (20*b^10*c*d + 21*a*b^9*d^2)/b^8)*x^2 + (80*b^10*c^2 + 340*a*b^9*c*d + 93*a^2
*b^8*d^2)/b^8)*x^2 + 5*(208*a*b^9*c^2 + 236*a^2*b^8*c*d + 3*a^3*b^7*d^2)/b^8)*x^2 + 15*(176*a^2*b^8*c^2 + 20*a
^3*b^7*c*d - 3*a^4*b^6*d^2)/b^8)*sqrt(b*x^2 + a)*x - 1/256*(80*a^3*b^2*c^2 - 20*a^4*b*c*d + 3*a^5*d^2)*log(abs
(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)

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maple [A]  time = 0.01, size = 308, normalized size = 1.28 \begin {gather*} \frac {3 a^{5} d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {5}{2}}}-\frac {5 a^{4} c d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{64 b^{\frac {3}{2}}}+\frac {5 a^{3} c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 \sqrt {b}}+\frac {3 \sqrt {b \,x^{2}+a}\, a^{4} d^{2} x}{256 b^{2}}-\frac {5 \sqrt {b \,x^{2}+a}\, a^{3} c d x}{64 b}+\frac {5 \sqrt {b \,x^{2}+a}\, a^{2} c^{2} x}{16}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3} d^{2} x}{128 b^{2}}-\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c d x}{96 b}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a \,c^{2} x}{24}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} d^{2} x^{3}}{10 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{2} d^{2} x}{160 b^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} a c d x}{24 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} c^{2} x}{6}-\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,d^{2} x}{80 b^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} c d x}{4 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)*(d*x^2+c)^2,x)

[Out]

1/10*d^2*x^3*(b*x^2+a)^(7/2)/b-3/80*d^2*a/b^2*x*(b*x^2+a)^(7/2)+1/160*d^2*a^2/b^2*x*(b*x^2+a)^(5/2)+1/128*d^2*
a^3/b^2*x*(b*x^2+a)^(3/2)+3/256*d^2*a^4/b^2*x*(b*x^2+a)^(1/2)+3/256*d^2*a^5/b^(5/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/
2))+1/4*c*d*x*(b*x^2+a)^(7/2)/b-1/24*c*d*a/b*x*(b*x^2+a)^(5/2)-5/96*c*d*a^2/b*x*(b*x^2+a)^(3/2)-5/64*c*d*a^3/b
*x*(b*x^2+a)^(1/2)-5/64*c*d*a^4/b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+1/6*c^2*x*(b*x^2+a)^(5/2)+5/24*c^2*a*x*(
b*x^2+a)^(3/2)+5/16*c^2*a^2*x*(b*x^2+a)^(1/2)+5/16*c^2*a^3/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))

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maxima [A]  time = 1.39, size = 286, normalized size = 1.19 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} d^{2} x^{3}}{10 \, b} + \frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2} x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a c^{2} x + \frac {5}{16} \, \sqrt {b x^{2} + a} a^{2} c^{2} x + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} c d x}{4 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a c d x}{24 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} c d x}{96 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a^{3} c d x}{64 \, b} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a d^{2} x}{80 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} d^{2} x}{160 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} d^{2} x}{128 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{4} d^{2} x}{256 \, b^{2}} + \frac {5 \, a^{3} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} - \frac {5 \, a^{4} c d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{64 \, b^{\frac {3}{2}}} + \frac {3 \, a^{5} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c)^2,x, algorithm="maxima")

[Out]

1/10*(b*x^2 + a)^(7/2)*d^2*x^3/b + 1/6*(b*x^2 + a)^(5/2)*c^2*x + 5/24*(b*x^2 + a)^(3/2)*a*c^2*x + 5/16*sqrt(b*
x^2 + a)*a^2*c^2*x + 1/4*(b*x^2 + a)^(7/2)*c*d*x/b - 1/24*(b*x^2 + a)^(5/2)*a*c*d*x/b - 5/96*(b*x^2 + a)^(3/2)
*a^2*c*d*x/b - 5/64*sqrt(b*x^2 + a)*a^3*c*d*x/b - 3/80*(b*x^2 + a)^(7/2)*a*d^2*x/b^2 + 1/160*(b*x^2 + a)^(5/2)
*a^2*d^2*x/b^2 + 1/128*(b*x^2 + a)^(3/2)*a^3*d^2*x/b^2 + 3/256*sqrt(b*x^2 + a)*a^4*d^2*x/b^2 + 5/16*a^3*c^2*ar
csinh(b*x/sqrt(a*b))/sqrt(b) - 5/64*a^4*c*d*arcsinh(b*x/sqrt(a*b))/b^(3/2) + 3/256*a^5*d^2*arcsinh(b*x/sqrt(a*
b))/b^(5/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(5/2)*(c + d*x^2)^2,x)

[Out]

int((a + b*x^2)^(5/2)*(c + d*x^2)^2, x)

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sympy [B]  time = 58.63, size = 537, normalized size = 2.23 \begin {gather*} - \frac {3 a^{\frac {9}{2}} d^{2} x}{256 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {7}{2}} c d x}{64 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{\frac {7}{2}} d^{2} x^{3}}{256 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {5}{2}} c^{2} x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {3 a^{\frac {5}{2}} c^{2} x}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {133 a^{\frac {5}{2}} c d x^{3}}{192 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {129 a^{\frac {5}{2}} d^{2} x^{5}}{640 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} b c^{2} x^{3}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {127 a^{\frac {3}{2}} b c d x^{5}}{96 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {73 a^{\frac {3}{2}} b d^{2} x^{7}}{160 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {17 \sqrt {a} b^{2} c^{2} x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {23 \sqrt {a} b^{2} c d x^{7}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {29 \sqrt {a} b^{2} d^{2} x^{9}}{80 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{5} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 b^{\frac {5}{2}}} - \frac {5 a^{4} c d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{64 b^{\frac {3}{2}}} + \frac {5 a^{3} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 \sqrt {b}} + \frac {b^{3} c^{2} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{3} c d x^{9}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{3} d^{2} x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)*(d*x**2+c)**2,x)

[Out]

-3*a**(9/2)*d**2*x/(256*b**2*sqrt(1 + b*x**2/a)) + 5*a**(7/2)*c*d*x/(64*b*sqrt(1 + b*x**2/a)) - a**(7/2)*d**2*
x**3/(256*b*sqrt(1 + b*x**2/a)) + a**(5/2)*c**2*x*sqrt(1 + b*x**2/a)/2 + 3*a**(5/2)*c**2*x/(16*sqrt(1 + b*x**2
/a)) + 133*a**(5/2)*c*d*x**3/(192*sqrt(1 + b*x**2/a)) + 129*a**(5/2)*d**2*x**5/(640*sqrt(1 + b*x**2/a)) + 35*a
**(3/2)*b*c**2*x**3/(48*sqrt(1 + b*x**2/a)) + 127*a**(3/2)*b*c*d*x**5/(96*sqrt(1 + b*x**2/a)) + 73*a**(3/2)*b*
d**2*x**7/(160*sqrt(1 + b*x**2/a)) + 17*sqrt(a)*b**2*c**2*x**5/(24*sqrt(1 + b*x**2/a)) + 23*sqrt(a)*b**2*c*d*x
**7/(24*sqrt(1 + b*x**2/a)) + 29*sqrt(a)*b**2*d**2*x**9/(80*sqrt(1 + b*x**2/a)) + 3*a**5*d**2*asinh(sqrt(b)*x/
sqrt(a))/(256*b**(5/2)) - 5*a**4*c*d*asinh(sqrt(b)*x/sqrt(a))/(64*b**(3/2)) + 5*a**3*c**2*asinh(sqrt(b)*x/sqrt
(a))/(16*sqrt(b)) + b**3*c**2*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a)) + b**3*c*d*x**9/(4*sqrt(a)*sqrt(1 + b*x**2/a
)) + b**3*d**2*x**11/(10*sqrt(a)*sqrt(1 + b*x**2/a))

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