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

Optimal. Leaf size=147 \[ \frac {2 (d+e x)^{11/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{11 e^5}+\frac {2 d^2 (d+e x)^{7/2} (c d-b e)^2}{7 e^5}-\frac {4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}-\frac {4 d (d+e x)^{9/2} (c d-b e) (2 c d-b e)}{9 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \]

[Out]

2/7*d^2*(-b*e+c*d)^2*(e*x+d)^(7/2)/e^5-4/9*d*(-b*e+c*d)*(-b*e+2*c*d)*(e*x+d)^(9/2)/e^5+2/11*(b^2*e^2-6*b*c*d*e
+6*c^2*d^2)*(e*x+d)^(11/2)/e^5-4/13*c*(-b*e+2*c*d)*(e*x+d)^(13/2)/e^5+2/15*c^2*(e*x+d)^(15/2)/e^5

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Rubi [A]  time = 0.07, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {698} \[ \frac {2 (d+e x)^{11/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{11 e^5}+\frac {2 d^2 (d+e x)^{7/2} (c d-b e)^2}{7 e^5}-\frac {4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}-\frac {4 d (d+e x)^{9/2} (c d-b e) (2 c d-b e)}{9 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(7/2))/(7*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^5) + (2*(
6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^5) +
(2*c^2*(d + e*x)^(15/2))/(15*e^5)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac {d^2 (c d-b e)^2 (d+e x)^{5/2}}{e^4}+\frac {2 d (c d-b e) (-2 c d+b e) (d+e x)^{7/2}}{e^4}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{11/2}}{e^4}+\frac {c^2 (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac {2 d^2 (c d-b e)^2 (d+e x)^{7/2}}{7 e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{9/2}}{9 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{11 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 124, normalized size = 0.84 \[ \frac {2 (d+e x)^{7/2} \left (65 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 b c e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(65*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 30*b*c*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^
2 + 231*e^3*x^3) + c^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*e^5
)

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fricas [A]  time = 0.82, size = 251, normalized size = 1.71 \[ \frac {2 \, {\left (3003 \, c^{2} e^{7} x^{7} + 128 \, c^{2} d^{7} - 480 \, b c d^{6} e + 520 \, b^{2} d^{5} e^{2} + 231 \, {\left (31 \, c^{2} d e^{6} + 30 \, b c e^{7}\right )} x^{6} + 63 \, {\left (71 \, c^{2} d^{2} e^{5} + 270 \, b c d e^{6} + 65 \, b^{2} e^{7}\right )} x^{5} + 35 \, {\left (c^{2} d^{3} e^{4} + 318 \, b c d^{2} e^{5} + 299 \, b^{2} d e^{6}\right )} x^{4} - 5 \, {\left (8 \, c^{2} d^{4} e^{3} - 30 \, b c d^{3} e^{4} - 1469 \, b^{2} d^{2} e^{5}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{5} e^{2} - 60 \, b c d^{4} e^{3} + 65 \, b^{2} d^{3} e^{4}\right )} x^{2} - 4 \, {\left (16 \, c^{2} d^{6} e - 60 \, b c d^{5} e^{2} + 65 \, b^{2} d^{4} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*c^2*e^7*x^7 + 128*c^2*d^7 - 480*b*c*d^6*e + 520*b^2*d^5*e^2 + 231*(31*c^2*d*e^6 + 30*b*c*e^7)*x^
6 + 63*(71*c^2*d^2*e^5 + 270*b*c*d*e^6 + 65*b^2*e^7)*x^5 + 35*(c^2*d^3*e^4 + 318*b*c*d^2*e^5 + 299*b^2*d*e^6)*
x^4 - 5*(8*c^2*d^4*e^3 - 30*b*c*d^3*e^4 - 1469*b^2*d^2*e^5)*x^3 + 3*(16*c^2*d^5*e^2 - 60*b*c*d^4*e^3 + 65*b^2*
d^3*e^4)*x^2 - 4*(16*c^2*d^6*e - 60*b*c*d^5*e^2 + 65*b^2*d^4*e^3)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.58, size = 916, normalized size = 6.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b^2*d^3*e^(-2) + 2574*(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)*b*c*d^3*e^(-3) + 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^3*e^(-4) + 3861*(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)*b^2*d^2*e^(-2) + 858*(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)*b*c*d^2*e^(-3) + 195*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 99
0*(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^2*e
^(-4) + 429*(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)*b^2*d*e^(-2) + 390*(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)*b*c*d*e^(-3) + 45*(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*d*e^(-4) + 65*(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)*b^2*e^(-2) + 30*(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)*b*c*e^(-3) +
7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 +
32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^
2*e^(-4))*e^(-1)

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maple [A]  time = 0.05, size = 141, normalized size = 0.96 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 c^{2} x^{4} e^{4}+6930 b c \,e^{4} x^{3}-1848 c^{2} d \,e^{3} x^{3}+4095 b^{2} e^{4} x^{2}-3780 b c d \,e^{3} x^{2}+1008 c^{2} d^{2} e^{2} x^{2}-1820 b^{2} d \,e^{3} x +1680 b c \,d^{2} e^{2} x -448 c^{2} d^{3} e x +520 b^{2} d^{2} e^{2}-480 b c \,d^{3} e +128 c^{2} d^{4}\right )}{45045 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(e*x+d)^(7/2)*(3003*c^2*e^4*x^4+6930*b*c*e^4*x^3-1848*c^2*d*e^3*x^3+4095*b^2*e^4*x^2-3780*b*c*d*e^3*x^
2+1008*c^2*d^2*e^2*x^2-1820*b^2*d*e^3*x+1680*b*c*d^2*e^2*x-448*c^2*d^3*e*x+520*b^2*d^2*e^2-480*b*c*d^3*e+128*c
^2*d^4)/e^5

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maxima [A]  time = 1.40, size = 139, normalized size = 0.95 \[ \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{2} - 6930 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 4095 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 10010 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^2 - 6930*(2*c^2*d - b*c*e)*(e*x + d)^(13/2) + 4095*(6*c^2*d^2 - 6*b*c*d*e + b
^2*e^2)*(e*x + d)^(11/2) - 10010*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(e*x + d)^(9/2) + 6435*(c^2*d^4 - 2*b*c
*d^3*e + b^2*d^2*e^2)*(e*x + d)^(7/2))/e^5

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mupad [B]  time = 0.04, size = 138, normalized size = 0.94 \[ \frac {2\,c^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}-\frac {{\left (d+e\,x\right )}^{9/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{9\,e^5}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{11\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*c^2*(d + e*x)^(15/2))/(15*e^5) - ((d + e*x)^(9/2)*(8*c^2*d^3 + 4*b^2*d*e^2 - 12*b*c*d^2*e))/(9*e^5) + ((d +
e*x)^(11/2)*(2*b^2*e^2 + 12*c^2*d^2 - 12*b*c*d*e))/(11*e^5) - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(13/2))/(13*e^5)
+ (2*d^2*(b*e - c*d)^2*(d + e*x)^(7/2))/(7*e^5)

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sympy [B]  time = 27.20, size = 695, normalized size = 4.73 \[ \frac {2 b^{2} d^{2} \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 b^{2} d \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 b^{2} \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^{3}} + \frac {4 b c d^{2} \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^{4}} + \frac {8 b c 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^{4}} + \frac {4 b c \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^{4}} + \frac {2 c^{2} d^{2} \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 {4 c^{2} d \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}} + \frac {2 c^{2} \left (\frac {d^{6} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {6 d^{5} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {15 d^{4} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {20 d^{3} \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {15 d^{2} \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {6 d \left (d + e x\right )^{\frac {13}{2}}}{13} + \frac {\left (d + e x\right )^{\frac {15}{2}}}{15}\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**2*d**2*(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*b**2*d*(-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*b**2*(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**3 + 4*b*c*d**2*(-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**4 + 8*b*c*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**4 + 4*b*c*(-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**4 + 2*c**2*d**2*(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 + 4*c**2*d*(-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 + 2*c**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3
*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5

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