3.6.19 \(\int (d+e x)^{3/2} (a+c x^2)^2 \, dx\)

Optimal. Leaf size=127 \[ \frac {4 c (d+e x)^{9/2} \left (a e^2+3 c d^2\right )}{9 e^5}-\frac {8 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^5}+\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5}-\frac {8 c^2 d (d+e x)^{11/2}}{11 e^5} \]

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Rubi [A]  time = 0.05, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {697} \begin {gather*} \frac {4 c (d+e x)^{9/2} \left (a e^2+3 c d^2\right )}{9 e^5}-\frac {8 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^5}+\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5}-\frac {8 c^2 d (d+e x)^{11/2}}{11 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)*(a + c*x^2)^2,x]

[Out]

(2*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2))/(5*e^5) - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)^(7/2))/(7*e^5) + (4*c*(3*c*d^
2 + a*e^2)*(d + e*x)^(9/2))/(9*e^5) - (8*c^2*d*(d + e*x)^(11/2))/(11*e^5) + (2*c^2*(d + e*x)^(13/2))/(13*e^5)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int (d+e x)^{3/2} \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^4}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^{5/2}}{e^4}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{7/2}}{e^4}-\frac {4 c^2 d (d+e x)^{9/2}}{e^4}+\frac {c^2 (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{5 e^5}-\frac {8 c d \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{9/2}}{9 e^5}-\frac {8 c^2 d (d+e x)^{11/2}}{11 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 97, normalized size = 0.76 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (9009 a^2 e^4+286 a c e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(9009*a^2*e^4 + 286*a*c*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 3*c^2*(128*d^4 - 320*d^3*e*x
+ 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(45045*e^5)

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IntegrateAlgebraic [A]  time = 0.06, size = 123, normalized size = 0.97 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (9009 a^2 e^4+18018 a c d^2 e^2-25740 a c d e^2 (d+e x)+10010 a c e^2 (d+e x)^2+9009 c^2 d^4-25740 c^2 d^3 (d+e x)+30030 c^2 d^2 (d+e x)^2-16380 c^2 d (d+e x)^3+3465 c^2 (d+e x)^4\right )}{45045 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(9009*c^2*d^4 + 18018*a*c*d^2*e^2 + 9009*a^2*e^4 - 25740*c^2*d^3*(d + e*x) - 25740*a*c*d*e^
2*(d + e*x) + 30030*c^2*d^2*(d + e*x)^2 + 10010*a*c*e^2*(d + e*x)^2 - 16380*c^2*d*(d + e*x)^3 + 3465*c^2*(d +
e*x)^4))/(45045*e^5)

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fricas [A]  time = 0.40, size = 181, normalized size = 1.43 \begin {gather*} \frac {2 \, {\left (3465 \, c^{2} e^{6} x^{6} + 4410 \, c^{2} d e^{5} x^{5} + 384 \, c^{2} d^{6} + 2288 \, a c d^{4} e^{2} + 9009 \, a^{2} d^{2} e^{4} + 35 \, {\left (3 \, c^{2} d^{2} e^{4} + 286 \, a c e^{6}\right )} x^{4} - 20 \, {\left (6 \, c^{2} d^{3} e^{3} - 715 \, a c d e^{5}\right )} x^{3} + 3 \, {\left (48 \, c^{2} d^{4} e^{2} + 286 \, a c d^{2} e^{4} + 3003 \, a^{2} e^{6}\right )} x^{2} - 2 \, {\left (96 \, c^{2} d^{5} e + 572 \, a c d^{3} e^{3} - 9009 \, a^{2} d e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*c^2*e^6*x^6 + 4410*c^2*d*e^5*x^5 + 384*c^2*d^6 + 2288*a*c*d^4*e^2 + 9009*a^2*d^2*e^4 + 35*(3*c^2
*d^2*e^4 + 286*a*c*e^6)*x^4 - 20*(6*c^2*d^3*e^3 - 715*a*c*d*e^5)*x^3 + 3*(48*c^2*d^4*e^2 + 286*a*c*d^2*e^4 + 3
003*a^2*e^6)*x^2 - 2*(96*c^2*d^5*e + 572*a*c*d^3*e^3 - 9009*a^2*d*e^5)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.20, size = 500, normalized size = 3.94 \begin {gather*} \frac {2}{45045} \, {\left (6006 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a c d^{2} e^{\left (-2\right )} + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c^{2} d^{2} e^{\left (-4\right )} + 5148 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a c d e^{\left (-2\right )} + 130 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} c^{2} d e^{\left (-4\right )} + 45045 \, \sqrt {x e + d} a^{2} d^{2} + 30030 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} d + 286 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a c e^{\left (-2\right )} + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c^{2} e^{\left (-4\right )} + 3003 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c*d^2*e^(-2) + 143*(35*(x*e
+ d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4
)*c^2*d^2*e^(-4) + 5148*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*
d^3)*a*c*d*e^(-2) + 130*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d
)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*c^2*d*e^(-4) + 45045*sqrt(x*e + d)*a^2*d^2 + 3
0030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*d + 286*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e
+ d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*c*e^(-2) + 15*(231*(x*e + d)^(13/2) - 1638
*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x
*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*c^2*e^(-4) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*s
qrt(x*e + d)*d^2)*a^2)*e^(-1)

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maple [A]  time = 0.05, size = 106, normalized size = 0.83 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3465 c^{2} x^{4} e^{4}-2520 c^{2} d \,x^{3} e^{3}+10010 a c \,e^{4} x^{2}+1680 c^{2} d^{2} e^{2} x^{2}-5720 a c d \,e^{3} x -960 c^{2} d^{3} e x +9009 a^{2} e^{4}+2288 a c \,d^{2} e^{2}+384 c^{2} d^{4}\right )}{45045 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(e*x+d)^(5/2)*(3465*c^2*e^4*x^4-2520*c^2*d*e^3*x^3+10010*a*c*e^4*x^2+1680*c^2*d^2*e^2*x^2-5720*a*c*d*e
^3*x-960*c^2*d^3*e*x+9009*a^2*e^4+2288*a*c*d^2*e^2+384*c^2*d^4)/e^5

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maxima [A]  time = 1.35, size = 113, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{2} - 16380 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{2} d + 10010 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 25740 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{45045 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*(e*x + d)^(13/2)*c^2 - 16380*(e*x + d)^(11/2)*c^2*d + 10010*(3*c^2*d^2 + a*c*e^2)*(e*x + d)^(9/2
) - 25740*(c^2*d^3 + a*c*d*e^2)*(e*x + d)^(7/2) + 9009*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^(5/2))/e^
5

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mupad [B]  time = 0.04, size = 114, normalized size = 0.90 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}-\frac {\left (8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {\left (12\,c^2\,d^2+4\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}-\frac {8\,c^2\,d\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2*(d + e*x)^(3/2),x)

[Out]

(2*c^2*(d + e*x)^(13/2))/(13*e^5) - ((8*c^2*d^3 + 8*a*c*d*e^2)*(d + e*x)^(7/2))/(7*e^5) + (2*(a*e^2 + c*d^2)^2
*(d + e*x)^(5/2))/(5*e^5) + ((12*c^2*d^2 + 4*a*c*e^2)*(d + e*x)^(9/2))/(9*e^5) - (8*c^2*d*(d + e*x)^(11/2))/(1
1*e^5)

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sympy [A]  time = 12.68, size = 328, normalized size = 2.58 \begin {gather*} a^{2} d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 a^{2} \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {4 a c d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {4 a c \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {2 c^{2} d \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} + \frac {2 c^{2} \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**2*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 4*a*c*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 4
*a*c*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3
+ 2*c**2*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**
(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 2*c**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d +
e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5

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