3.5.6 \(\int \frac {(a+b x^2)^2 (c+d x^2)^2}{x^{7/2}} \, dx\) [406]

Optimal. Leaf size=95 \[ -\frac {2 a^2 c^2}{5 x^{5/2}}-\frac {4 a c (b c+a d)}{\sqrt {x}}+\frac {2}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{3/2}+\frac {4}{7} b d (b c+a d) x^{7/2}+\frac {2}{11} b^2 d^2 x^{11/2} \]

[Out]

-2/5*a^2*c^2/x^(5/2)+2/3*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^(3/2)+4/7*b*d*(a*d+b*c)*x^(7/2)+2/11*b^2*d^2*x^(11/2)-4
*a*c*(a*d+b*c)/x^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {459} \begin {gather*} \frac {2}{3} x^{3/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {2 a^2 c^2}{5 x^{5/2}}+\frac {4}{7} b d x^{7/2} (a d+b c)-\frac {4 a c (a d+b c)}{\sqrt {x}}+\frac {2}{11} b^2 d^2 x^{11/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2)^2)/x^(7/2),x]

[Out]

(-2*a^2*c^2)/(5*x^(5/2)) - (4*a*c*(b*c + a*d))/Sqrt[x] + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(3/2))/3 + (4*b*
d*(b*c + a*d)*x^(7/2))/7 + (2*b^2*d^2*x^(11/2))/11

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{7/2}} \, dx &=\int \left (\frac {a^2 c^2}{x^{7/2}}+\frac {2 a c (b c+a d)}{x^{3/2}}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) \sqrt {x}+2 b d (b c+a d) x^{5/2}+b^2 d^2 x^{9/2}\right ) \, dx\\ &=-\frac {2 a^2 c^2}{5 x^{5/2}}-\frac {4 a c (b c+a d)}{\sqrt {x}}+\frac {2}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{3/2}+\frac {4}{7} b d (b c+a d) x^{7/2}+\frac {2}{11} b^2 d^2 x^{11/2}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 93, normalized size = 0.98 \begin {gather*} \frac {-154 a^2 \left (3 c^2+30 c d x^2-5 d^2 x^4\right )+220 a b x^2 \left (-21 c^2+14 c d x^2+3 d^2 x^4\right )+10 b^2 x^4 \left (77 c^2+66 c d x^2+21 d^2 x^4\right )}{1155 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^(7/2),x]

[Out]

(-154*a^2*(3*c^2 + 30*c*d*x^2 - 5*d^2*x^4) + 220*a*b*x^2*(-21*c^2 + 14*c*d*x^2 + 3*d^2*x^4) + 10*b^2*x^4*(77*c
^2 + 66*c*d*x^2 + 21*d^2*x^4))/(1155*x^(5/2))

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Maple [A]
time = 0.09, size = 89, normalized size = 0.94

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {4 a b \,d^{2} x^{\frac {7}{2}}}{7}+\frac {4 b^{2} c d \,x^{\frac {7}{2}}}{7}+\frac {2 a^{2} d^{2} x^{\frac {3}{2}}}{3}+\frac {8 a b c d \,x^{\frac {3}{2}}}{3}+\frac {2 b^{2} c^{2} x^{\frac {3}{2}}}{3}-\frac {2 a^{2} c^{2}}{5 x^{\frac {5}{2}}}-\frac {4 a c \left (a d +b c \right )}{\sqrt {x}}\)	\(89\)
default	\(\frac {2 b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {4 a b \,d^{2} x^{\frac {7}{2}}}{7}+\frac {4 b^{2} c d \,x^{\frac {7}{2}}}{7}+\frac {2 a^{2} d^{2} x^{\frac {3}{2}}}{3}+\frac {8 a b c d \,x^{\frac {3}{2}}}{3}+\frac {2 b^{2} c^{2} x^{\frac {3}{2}}}{3}-\frac {2 a^{2} c^{2}}{5 x^{\frac {5}{2}}}-\frac {4 a c \left (a d +b c \right )}{\sqrt {x}}\)	\(89\)
gosper	\(-\frac {2 \left (-105 b^{2} d^{2} x^{8}-330 a b \,d^{2} x^{6}-330 b^{2} c d \,x^{6}-385 a^{2} d^{2} x^{4}-1540 a b c d \,x^{4}-385 b^{2} c^{2} x^{4}+2310 a^{2} c d \,x^{2}+2310 a b \,c^{2} x^{2}+231 a^{2} c^{2}\right )}{1155 x^{\frac {5}{2}}}\)	\(97\)
trager	\(-\frac {2 \left (-105 b^{2} d^{2} x^{8}-330 a b \,d^{2} x^{6}-330 b^{2} c d \,x^{6}-385 a^{2} d^{2} x^{4}-1540 a b c d \,x^{4}-385 b^{2} c^{2} x^{4}+2310 a^{2} c d \,x^{2}+2310 a b \,c^{2} x^{2}+231 a^{2} c^{2}\right )}{1155 x^{\frac {5}{2}}}\)	\(97\)
risch	\(-\frac {2 \left (-105 b^{2} d^{2} x^{8}-330 a b \,d^{2} x^{6}-330 b^{2} c d \,x^{6}-385 a^{2} d^{2} x^{4}-1540 a b c d \,x^{4}-385 b^{2} c^{2} x^{4}+2310 a^{2} c d \,x^{2}+2310 a b \,c^{2} x^{2}+231 a^{2} c^{2}\right )}{1155 x^{\frac {5}{2}}}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^2/x^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/11*b^2*d^2*x^(11/2)+4/7*a*b*d^2*x^(7/2)+4/7*b^2*c*d*x^(7/2)+2/3*a^2*d^2*x^(3/2)+8/3*a*b*c*d*x^(3/2)+2/3*b^2*
c^2*x^(3/2)-2/5*a^2*c^2/x^(5/2)-4*a*c*(a*d+b*c)/x^(1/2)

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Maxima [A]
time = 0.28, size = 87, normalized size = 0.92 \begin {gather*} \frac {2}{11} \, b^{2} d^{2} x^{\frac {11}{2}} + \frac {4}{7} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {7}{2}} + \frac {2}{3} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {3}{2}} - \frac {2 \, {\left (a^{2} c^{2} + 10 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{5 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2/x^(7/2),x, algorithm="maxima")

[Out]

2/11*b^2*d^2*x^(11/2) + 4/7*(b^2*c*d + a*b*d^2)*x^(7/2) + 2/3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(3/2) - 2/5*(a
^2*c^2 + 10*(a*b*c^2 + a^2*c*d)*x^2)/x^(5/2)

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Fricas [A]
time = 0.60, size = 87, normalized size = 0.92 \begin {gather*} \frac {2 \, {\left (105 \, b^{2} d^{2} x^{8} + 330 \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + 385 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 231 \, a^{2} c^{2} - 2310 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{1155 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2/x^(7/2),x, algorithm="fricas")

[Out]

2/1155*(105*b^2*d^2*x^8 + 330*(b^2*c*d + a*b*d^2)*x^6 + 385*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 - 231*a^2*c^2
- 2310*(a*b*c^2 + a^2*c*d)*x^2)/x^(5/2)

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Sympy [A]
time = 0.93, size = 133, normalized size = 1.40 \begin {gather*} - \frac {2 a^{2} c^{2}}{5 x^{\frac {5}{2}}} - \frac {4 a^{2} c d}{\sqrt {x}} + \frac {2 a^{2} d^{2} x^{\frac {3}{2}}}{3} - \frac {4 a b c^{2}}{\sqrt {x}} + \frac {8 a b c d x^{\frac {3}{2}}}{3} + \frac {4 a b d^{2} x^{\frac {7}{2}}}{7} + \frac {2 b^{2} c^{2} x^{\frac {3}{2}}}{3} + \frac {4 b^{2} c d x^{\frac {7}{2}}}{7} + \frac {2 b^{2} d^{2} x^{\frac {11}{2}}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**2/x**(7/2),x)

[Out]

-2*a**2*c**2/(5*x**(5/2)) - 4*a**2*c*d/sqrt(x) + 2*a**2*d**2*x**(3/2)/3 - 4*a*b*c**2/sqrt(x) + 8*a*b*c*d*x**(3
/2)/3 + 4*a*b*d**2*x**(7/2)/7 + 2*b**2*c**2*x**(3/2)/3 + 4*b**2*c*d*x**(7/2)/7 + 2*b**2*d**2*x**(11/2)/11

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Giac [A]
time = 0.64, size = 96, normalized size = 1.01 \begin {gather*} \frac {2}{11} \, b^{2} d^{2} x^{\frac {11}{2}} + \frac {4}{7} \, b^{2} c d x^{\frac {7}{2}} + \frac {4}{7} \, a b d^{2} x^{\frac {7}{2}} + \frac {2}{3} \, b^{2} c^{2} x^{\frac {3}{2}} + \frac {8}{3} \, a b c d x^{\frac {3}{2}} + \frac {2}{3} \, a^{2} d^{2} x^{\frac {3}{2}} - \frac {2 \, {\left (10 \, a b c^{2} x^{2} + 10 \, a^{2} c d x^{2} + a^{2} c^{2}\right )}}{5 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2/x^(7/2),x, algorithm="giac")

[Out]

2/11*b^2*d^2*x^(11/2) + 4/7*b^2*c*d*x^(7/2) + 4/7*a*b*d^2*x^(7/2) + 2/3*b^2*c^2*x^(3/2) + 8/3*a*b*c*d*x^(3/2)
+ 2/3*a^2*d^2*x^(3/2) - 2/5*(10*a*b*c^2*x^2 + 10*a^2*c*d*x^2 + a^2*c^2)/x^(5/2)

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Mupad [B]
time = 0.03, size = 86, normalized size = 0.91 \begin {gather*} x^{3/2}\,\left (\frac {2\,a^2\,d^2}{3}+\frac {8\,a\,b\,c\,d}{3}+\frac {2\,b^2\,c^2}{3}\right )-\frac {x^2\,\left (4\,d\,a^2\,c+4\,b\,a\,c^2\right )+\frac {2\,a^2\,c^2}{5}}{x^{5/2}}+\frac {2\,b^2\,d^2\,x^{11/2}}{11}+\frac {4\,b\,d\,x^{7/2}\,\left (a\,d+b\,c\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^2*(c + d*x^2)^2)/x^(7/2),x)

[Out]

x^(3/2)*((2*a^2*d^2)/3 + (2*b^2*c^2)/3 + (8*a*b*c*d)/3) - (x^2*(4*a*b*c^2 + 4*a^2*c*d) + (2*a^2*c^2)/5)/x^(5/2
) + (2*b^2*d^2*x^(11/2))/11 + (4*b*d*x^(7/2)*(a*d + b*c))/7

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